Number of ways in which an examiner can assign 30 marks to 8 questions, giving not less than 2 marks

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6.Permutation and Combination

hard

The number of ways in which an examiner can assign $30$ marks to $8$ questions, giving not less than $2$ marks to any question, is

$^{30}{C_7}$

$^{21}{C_8}$

$^{21}{C_7}$

$^{30}{C_8}$

(JEE MAIN-2013)

$30$ marks to be alloted to $8$ question. Each question has to be given $ \geqslant 2$ marks

Let questions be $a,b,c,d,e,f,g,h$ and $a+b+c+d+e+f+g+h=30$

Let $a=a_1+2$ so, ${a_1} \geqslant 0$

$b=a_2+2$ so, ${a_2} \geqslant 0,………{a_8} \geqslant 0$

So, $\left. \begin{gathered}
{a_1} + {a_2} + ……… + {a_8} \hfill \\
+ 2 + 2 + ………. + 2 \hfill \\
\end{gathered} \right\} = 30$

$ \Rightarrow \,{a_1} + {a_2} + ……. + {a_8} = 30 – 16 = 14$

So, this is a problem of distributing $14$ articles in $8$ groups.

Number of ways ${ = ^{14 + 8 – 1}}{C_{8 – 1}} = 21{C_7}$

Standard 11

Mathematics

easy

medium

normal

hard

(JEE MAIN-2024)

normal